curve #3988

y² + xy = x³ − 81556059719069376657064933643743504604764970x + 197488548627331729353041658652962519450921684418438094943084756900

a-invariants: [1, 0, 0, -81556059719069376657064933643743504604764970, 197488548627331729353041658652962519450921684418438094943084756900]
c-invariants: [3914690866515330079539116814899688221028718561, -170630106014014614161033865112459389800715644012752863563156773039441]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 15662834656502411939691591195371771384959399565490
discriminant (Δ): 17868772826149492413880920722801078677204878195067534446780445312238425425762049894273464111586229241627824859966937063217412900000000
Faltings height: 24.2742
naive height: 314.9432
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 37, 43, 53, 59, 83, 107, 109, 127, 157, 269, 347, 607, 3181, 1307863, 23637023
regulator: 96401226198752026.355559829738262060691642321011532628117840365
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:40:24 UTC
last updated: 2026-10-05 04:40:24 UTC

Witness: 12 independent points log in to add more points →

(-120243839912260957659297104/12321, 345750160067975373571358154414416131382/1367631)
(-9149881806292321788756944/1681, 47756914164329357832768619045499869622/68921)
(-9894134912029735099986933822669584421140/1052879890435813449, 395540011125440075111601861839739870240201612268539150849190/1080359378574418629790852293)
(-24618748843797520631413460/17161, 1254778192942578608982910260951296320730/2248091)
(34405472329478745822660306776114140/3024924471361, 4529688046080846164029387307614749620609577446500130/5261042413249663391)
(55239872304549819810969610/5329, -265536992712536667944395299817543706390/389017)
(5432796162540409927662977201205940/26396175961, 400056724053577608363792017848071386799688745340670/4288560312207709)
(-35944524506507656143908971470343814837/4549359237327844, -181180467181511079827903418405954538205573655648985130933/306849449138253035629672)
(24280430854176707767360492686484525/9498156920464, -2211854875208229930370365989520410676251196457045885/29272445798433365312)
(5958350020709394158783059923057187660/486553467013849, -10919794188130612066332454207937225348448069471811817810/10732375453592399646493)
(-10455797182038095222544549615854331148340/1095206155301034529, -373099400945104803573566481209876749365601522302353983468210/1146156215662052384399814833)
(350569198934455966970128918194403612068358601553699309953563859518685813990346197802034622849442040/45462698592138874554109501744925311817756996850086265813145370092129400484921, -1596219414013135361153929959436165225551413053066706904738545146935255967937114893999580669876663778347353041064702868865346424042800718206995182770/9693549151608346865941089511217369003021627158487330471590471507494674456853303380951023106451947660828233460396781)

Commentary

Unlisted rational-isogeny neighbor of curve#3705 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 12: 12 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:40:25 UTC · history